I often hear this claim. But looking up the Jewish diaspora on Wikipedia, there was the claim that that the Ashkenazi Jews who live mostly in Christian societies made up 92% of the Jewish population worldwide in 1931. It seems really strange to me that both claims can be true.
First, I'd like to suggest this answer, and this one about what persecution of Jews in Muslim countries actually looked like.
The premise of your question seems to be flawed, I think. The reason Ashkenazi Jews made up a huge share of world Jewry pre-Holocaust was that they'd been growing extremely quickly, not that Jews were moving there to avoid persecution. Jews moved to Eastern Europe largely to avoid persecution, but that was several centuries before the 20th century.
In fact, it was pretty rare for Jews to move from Muslim to Christian countries to avoid persecution. It happened a bit when fleeing the Almohads (a particularly intolerant Muslim government in the 12th century, in North Africa and Iberia), but the much larger example is Jews fleeing the Inquisition in Spain fleeing to North Africa, Turkey, and the Balkans, all of which were Muslim-controlled. But, many fled to other Christian countries too (and from the Almohads to other Muslim countries).
The issue with answering is this is that it's correct that Jewish communities were persecuted less, but it's easy to make false generalizations. It's easy to read about a place where there was a pogrom or an expulsion every 100 years and mentally compress that time frame, when in reality there would be a few generations of relative acceptance.
But on the whole, it's largely correct. During the Middle Ages many Christian societies eroded the protection and communal autonomy for Jews that was inherited from the Roman Empire. There were occasional expulsions, massacres, punitive taxes, and legal disabilities. While those existed to an extent in Muslim countries (especially the taxes and legal disabilities part), it was much more common for an entire Jewish community to be kicked out of a city or region, or to forcibly be converted, in Christendom than under Islam.
That shouldn't be taken to mean that everything was harmonious and happy, as I noted before, and in the linked comments above. With regard to the specific wording of your question, one can't just ignore history after 1900 when looking at the historical treatment of Jews in Muslim countries. But taken on the whole, government policy towards Jews in Muslim countries was generally less oppressive than in Christian ones.